statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
lt_or_le_of_mul_le_mul [covariant_class α α (*) (<)]
[covariant_class α α (swap (*)) (≤)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂ | by { contrapose!, exact λ h, mul_lt_mul_of_le_of_lt h.1 h.2 } | lemma | lt_or_le_of_mul_le_mul | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [
"covariant_class",
"mul_lt_mul_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_or_le_of_mul_le_mul [covariant_class α α (*) (<)]
[covariant_class α α (swap (*)) (<)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂ | by { contrapose!, exact λ h, mul_lt_mul_of_lt_of_lt h.1 h.2 } | lemma | le_or_le_of_mul_le_mul | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [
"covariant_class",
"mul_lt_mul_of_lt_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_iff_of_le_of_le [covariant_class α α (*) (≤)]
[covariant_class α α (swap (*)) (≤)] [covariant_class α α (*) (<)]
[covariant_class α α (swap (*)) (<)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) :
a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂ | begin
refine ⟨lt_or_lt_of_mul_lt_mul, _⟩,
rintro (ha | hb),
{ exact mul_lt_mul_of_lt_of_le ha hb },
{ exact mul_lt_mul_of_le_of_lt ha hb }
end | lemma | mul_lt_mul_iff_of_le_of_le | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [
"covariant_class",
"mul_lt_mul_of_le_of_lt",
"mul_lt_mul_of_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_le_mul_of_one_le_right [covariant_class α α (*) (≤)] {a b : α} (hb : 1 ≤ b) :
min a b ≤ a * b | min_le_iff.2 $ or.inl $ le_mul_of_one_le_right' hb | lemma | min_le_mul_of_one_le_right | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [
"covariant_class",
"le_mul_of_one_le_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_le_mul_of_one_le_left [covariant_class α α (function.swap (*)) (≤)] {a b : α}
(ha : 1 ≤ a) : min a b ≤ a * b | min_le_iff.2 $ or.inr $ le_mul_of_one_le_left' ha | lemma | min_le_mul_of_one_le_left | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [
"covariant_class",
"le_mul_of_one_le_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_le_mul_of_one_le [covariant_class α α (*) (≤)]
[covariant_class α α (function.swap (*)) (≤)] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
max a b ≤ a * b | max_le_iff.2 ⟨le_mul_of_one_le_right' hb, le_mul_of_one_le_left' ha⟩ | lemma | max_le_mul_of_one_le | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [
"covariant_class",
"le_mul_of_one_le_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_add_one [has_one α] [add_zero_class α] [partial_order α] [zero_le_one_class α]
[ne_zero (1 : α)] [covariant_class α α (+) (<)] (a : α) : a < a + 1 | lt_add_of_pos_right _ zero_lt_one | lemma | lt_add_one | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"add_zero_class",
"covariant_class",
"ne_zero",
"zero_le_one_class",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_one_add [has_one α] [add_zero_class α] [partial_order α] [zero_le_one_class α]
[ne_zero (1 : α)] [covariant_class α α (swap (+)) (<)] (a : α) : a < 1 + a | lt_add_of_pos_left _ zero_lt_one | lemma | lt_one_add | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"add_zero_class",
"covariant_class",
"ne_zero",
"zero_le_one_class",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_two [preorder α] [zero_le_one_class α] [covariant_class α α (+) (≤)] :
(0 : α) ≤ 2 | add_nonneg zero_le_one zero_le_one | lemma | zero_le_two | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"covariant_class",
"zero_le_one",
"zero_le_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_three [preorder α] [zero_le_one_class α] [covariant_class α α (+) (≤)] :
(0 : α) ≤ 3 | add_nonneg zero_le_two zero_le_one | lemma | zero_le_three | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"covariant_class",
"zero_le_one",
"zero_le_one_class",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_four [preorder α] [zero_le_one_class α] [covariant_class α α (+) (≤)] :
(0 : α) ≤ 4 | add_nonneg zero_le_two zero_le_two | lemma | zero_le_four | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"covariant_class",
"zero_le_one_class",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_two [has_le α] [zero_le_one_class α] [covariant_class α α (+) (≤)] :
(1 : α) ≤ 2 | calc 1 = 1 + 0 : (add_zero 1).symm
... ≤ 1 + 1 : add_le_add_left zero_le_one _ | lemma | one_le_two | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"covariant_class",
"zero_le_one",
"zero_le_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_two' [has_le α] [zero_le_one_class α] [covariant_class α α (swap (+)) (≤)] :
(1 : α) ≤ 2 | calc 1 = 0 + 1 : (zero_add 1).symm
... ≤ 1 + 1 : add_le_add_right zero_le_one _ | lemma | one_le_two' | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"covariant_class",
"zero_le_one",
"zero_le_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_two : (0 : α) < 2 | zero_lt_one.trans_le one_le_two | lemma | zero_lt_two | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"one_le_two"
] | See `zero_lt_two'` for a version with the type explicit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_lt_three : (0 : α) < 3 | lt_add_of_lt_of_nonneg zero_lt_two zero_le_one | lemma | zero_lt_three | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"zero_le_one",
"zero_lt_two"
] | See `zero_lt_three'` for a version with the type explicit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_lt_four : (0 : α) < 4 | lt_add_of_lt_of_nonneg zero_lt_two zero_le_two | lemma | zero_lt_four | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"zero_le_two",
"zero_lt_two"
] | See `zero_lt_four'` for a version with the type explicit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_lt_two' : (0 : α) < 2 | zero_lt_two | lemma | zero_lt_two' | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"zero_lt_two"
] | See `zero_lt_two` for a version with the type implicit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_lt_three' : (0 : α) < 3 | zero_lt_three | lemma | zero_lt_three' | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"zero_lt_three"
] | See `zero_lt_three` for a version with the type implicit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_lt_four' : (0 : α) < 4 | zero_lt_four | lemma | zero_lt_four' | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"zero_lt_four"
] | See `zero_lt_four` for a version with the type implicit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_le_one_class.ne_zero.two : ne_zero (2 : α) | ⟨zero_lt_two.ne'⟩ | instance | zero_le_one_class.ne_zero.two | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_one_class.ne_zero.three : ne_zero (3 : α) | ⟨zero_lt_three.ne'⟩ | instance | zero_le_one_class.ne_zero.three | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_one_class.ne_zero.four : ne_zero (4 : α) | ⟨zero_lt_four.ne'⟩ | instance | zero_le_one_class.ne_zero.four | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_two [covariant_class α α (+) (<)] : (1 : α) < 2 | lt_add_one _ | lemma | one_lt_two | algebra.order.monoid | src/algebra/order/monoid/nat_cast.lean | [
"algebra.order.monoid.lemmas",
"algebra.order.zero_le_one",
"data.nat.cast.defs"
] | [
"covariant_class",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_mul_le [has_le α] [has_mul α] [c : contravariant_class α α (*) (≤)] :
contravariant_class αᵒᵈ αᵒᵈ (*) (≤) | ⟨c.1.flip⟩ | instance | order_dual.contravariant_class_mul_le | algebra.order.monoid | src/algebra/order/monoid/order_dual.lean | [
"algebra.group.order_synonym",
"algebra.order.monoid.cancel.defs"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_mul_le [has_le α] [has_mul α] [c : covariant_class α α (*) (≤)] :
covariant_class αᵒᵈ αᵒᵈ (*) (≤) | ⟨c.1.flip⟩ | instance | order_dual.covariant_class_mul_le | algebra.order.monoid | src/algebra/order/monoid/order_dual.lean | [
"algebra.group.order_synonym",
"algebra.order.monoid.cancel.defs"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_swap_mul_le [has_le α] [has_mul α]
[c : contravariant_class α α (swap (*)) (≤)] :
contravariant_class αᵒᵈ αᵒᵈ (swap (*)) (≤) | ⟨c.1.flip⟩ | instance | order_dual.contravariant_class_swap_mul_le | algebra.order.monoid | src/algebra/order/monoid/order_dual.lean | [
"algebra.group.order_synonym",
"algebra.order.monoid.cancel.defs"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_swap_mul_le [has_le α] [has_mul α]
[c : covariant_class α α (swap (*)) (≤)] :
covariant_class αᵒᵈ αᵒᵈ (swap (*)) (≤) | ⟨c.1.flip⟩ | instance | order_dual.covariant_class_swap_mul_le | algebra.order.monoid | src/algebra/order/monoid/order_dual.lean | [
"algebra.group.order_synonym",
"algebra.order.monoid.cancel.defs"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_mul_lt [has_lt α] [has_mul α] [c : contravariant_class α α (*) (<)] :
contravariant_class αᵒᵈ αᵒᵈ (*) (<) | ⟨c.1.flip⟩ | instance | order_dual.contravariant_class_mul_lt | algebra.order.monoid | src/algebra/order/monoid/order_dual.lean | [
"algebra.group.order_synonym",
"algebra.order.monoid.cancel.defs"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_mul_lt [has_lt α] [has_mul α] [c : covariant_class α α (*) (<)] :
covariant_class αᵒᵈ αᵒᵈ (*) (<) | ⟨c.1.flip⟩ | instance | order_dual.covariant_class_mul_lt | algebra.order.monoid | src/algebra/order/monoid/order_dual.lean | [
"algebra.group.order_synonym",
"algebra.order.monoid.cancel.defs"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_swap_mul_lt [has_lt α] [has_mul α]
[c : contravariant_class α α (swap (*)) (<)] :
contravariant_class αᵒᵈ αᵒᵈ (swap (*)) (<) | ⟨c.1.flip⟩ | instance | order_dual.contravariant_class_swap_mul_lt | algebra.order.monoid | src/algebra/order/monoid/order_dual.lean | [
"algebra.group.order_synonym",
"algebra.order.monoid.cancel.defs"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_swap_mul_lt [has_lt α] [has_mul α]
[c : covariant_class α α (swap (*)) (<)] :
covariant_class αᵒᵈ αᵒᵈ (swap (*)) (<) | ⟨c.1.flip⟩ | instance | order_dual.covariant_class_swap_mul_lt | algebra.order.monoid | src/algebra/order/monoid/order_dual.lean | [
"algebra.group.order_synonym",
"algebra.order.monoid.cancel.defs"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_cancel_comm_monoid.to_contravariant_class [ordered_cancel_comm_monoid α] :
contravariant_class αᵒᵈ αᵒᵈ has_mul.mul has_le.le | { elim := λ a b c, ordered_cancel_comm_monoid.le_of_mul_le_mul_left a c b } | instance | order_dual.ordered_cancel_comm_monoid.to_contravariant_class | algebra.order.monoid | src/algebra/order/monoid/order_dual.lean | [
"algebra.group.order_synonym",
"algebra.order.monoid.cancel.defs"
] | [
"contravariant_class",
"ordered_cancel_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_bot : with_zero (multiplicative α) ≃* multiplicative (with_bot α) | by exact mul_equiv.refl _ | def | with_zero.to_mul_bot | algebra.order.monoid | src/algebra/order/monoid/to_mul_bot.lean | [
"algebra.order.with_zero",
"algebra.order.monoid.with_top",
"algebra.order.monoid.type_tags"
] | [
"mul_equiv.refl",
"multiplicative",
"with_bot"
] | Making an additive monoid multiplicative then adding a zero is the same as adding a bottom
element then making it multiplicative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_mul_bot_zero :
to_mul_bot (0 : with_zero (multiplicative α)) = multiplicative.of_add ⊥ | rfl | lemma | with_zero.to_mul_bot_zero | algebra.order.monoid | src/algebra/order/monoid/to_mul_bot.lean | [
"algebra.order.with_zero",
"algebra.order.monoid.with_top",
"algebra.order.monoid.type_tags"
] | [
"multiplicative",
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_bot_coe (x : multiplicative α) :
to_mul_bot ↑x = multiplicative.of_add (x.to_add : with_bot α) | rfl | lemma | with_zero.to_mul_bot_coe | algebra.order.monoid | src/algebra/order/monoid/to_mul_bot.lean | [
"algebra.order.with_zero",
"algebra.order.monoid.with_top",
"algebra.order.monoid.type_tags"
] | [
"multiplicative",
"multiplicative.of_add",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_bot_symm_bot :
to_mul_bot.symm (multiplicative.of_add (⊥ : with_bot α)) = 0 | rfl | lemma | with_zero.to_mul_bot_symm_bot | algebra.order.monoid | src/algebra/order/monoid/to_mul_bot.lean | [
"algebra.order.with_zero",
"algebra.order.monoid.with_top",
"algebra.order.monoid.type_tags"
] | [
"multiplicative.of_add",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_bot_coe_of_add (x : α) :
to_mul_bot.symm (multiplicative.of_add (x : with_bot α)) = multiplicative.of_add x | rfl | lemma | with_zero.to_mul_bot_coe_of_add | algebra.order.monoid | src/algebra/order/monoid/to_mul_bot.lean | [
"algebra.order.with_zero",
"algebra.order.monoid.with_top",
"algebra.order.monoid.type_tags"
] | [
"multiplicative.of_add",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_bot_strict_mono : strict_mono (@to_mul_bot α _) | λ x y, id | lemma | with_zero.to_mul_bot_strict_mono | algebra.order.monoid | src/algebra/order/monoid/to_mul_bot.lean | [
"algebra.order.with_zero",
"algebra.order.monoid.with_top",
"algebra.order.monoid.type_tags"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_bot_le : to_mul_bot a ≤ to_mul_bot b ↔ a ≤ b | iff.rfl | lemma | with_zero.to_mul_bot_le | algebra.order.monoid | src/algebra/order/monoid/to_mul_bot.lean | [
"algebra.order.with_zero",
"algebra.order.monoid.with_top",
"algebra.order.monoid.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_bot_lt : to_mul_bot a < to_mul_bot b ↔ a < b | iff.rfl | lemma | with_zero.to_mul_bot_lt | algebra.order.monoid | src/algebra/order/monoid/to_mul_bot.lean | [
"algebra.order.with_zero",
"algebra.order.monoid.with_top",
"algebra.order.monoid.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_le {a b : α} : of_mul a ≤ of_mul b ↔ a ≤ b | iff.rfl | lemma | additive.of_mul_le | algebra.order.monoid | src/algebra/order/monoid/type_tags.lean | [
"algebra.group.type_tags",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_lt {a b : α} : of_mul a < of_mul b ↔ a < b | iff.rfl | lemma | additive.of_mul_lt | algebra.order.monoid | src/algebra/order/monoid/type_tags.lean | [
"algebra.group.type_tags",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_le {a b : additive α} : to_mul a ≤ to_mul b ↔ a ≤ b | iff.rfl | lemma | additive.to_mul_le | algebra.order.monoid | src/algebra/order/monoid/type_tags.lean | [
"algebra.group.type_tags",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_lt {a b : additive α} : to_mul a < to_mul b ↔ a < b | iff.rfl | lemma | additive.to_mul_lt | algebra.order.monoid | src/algebra/order/monoid/type_tags.lean | [
"algebra.group.type_tags",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_add_le {a b : α} : of_add a ≤ of_add b ↔ a ≤ b | iff.rfl | lemma | multiplicative.of_add_le | algebra.order.monoid | src/algebra/order/monoid/type_tags.lean | [
"algebra.group.type_tags",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_add_lt {a b : α} : of_add a < of_add b ↔ a < b | iff.rfl | lemma | multiplicative.of_add_lt | algebra.order.monoid | src/algebra/order/monoid/type_tags.lean | [
"algebra.group.type_tags",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_le {a b : multiplicative α} : to_add a ≤ to_add b ↔ a ≤ b | iff.rfl | lemma | multiplicative.to_add_le | algebra.order.monoid | src/algebra/order/monoid/type_tags.lean | [
"algebra.group.type_tags",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs"
] | [
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_lt {a b : multiplicative α} : to_add a < to_add b ↔ a < b | iff.rfl | lemma | multiplicative.to_add_lt | algebra.order.monoid | src/algebra/order/monoid/type_tags.lean | [
"algebra.group.type_tags",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs"
] | [
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_coe [monoid α] [preorder α] {a b : αˣ} :
(a : α) ≤ b ↔ a ≤ b | iff.rfl | theorem | units.coe_le_coe | algebra.order.monoid | src/algebra/order/monoid/units.lean | [
"order.hom.basic",
"order.min_max",
"algebra.group.units"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lt_coe [monoid α] [preorder α] {a b : αˣ} :
(a : α) < b ↔ a < b | iff.rfl | theorem | units.coe_lt_coe | algebra.order.monoid | src/algebra/order/monoid/units.lean | [
"order.hom.basic",
"order.min_max",
"algebra.group.units"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_embedding_coe [monoid α] [linear_order α] : αˣ ↪o α | ⟨⟨coe, ext⟩, λ _ _, iff.rfl⟩ | def | units.order_embedding_coe | algebra.order.monoid | src/algebra/order/monoid/units.lean | [
"order.hom.basic",
"order.min_max",
"algebra.group.units"
] | [
"monoid"
] | `coe : αˣ → α` as an order embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
max_coe [monoid α] [linear_order α] {a b : αˣ} :
(↑(max a b) : α) = max a b | monotone.map_max order_embedding_coe.monotone | theorem | units.max_coe | algebra.order.monoid | src/algebra/order/monoid/units.lean | [
"order.hom.basic",
"order.min_max",
"algebra.group.units"
] | [
"monoid",
"monotone.map_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_coe [monoid α] [linear_order α] {a b : αˣ} :
(↑(min a b) : α) = min a b | monotone.map_min order_embedding_coe.monotone | theorem | units.min_coe | algebra.order.monoid | src/algebra/order/monoid/units.lean | [
"order.hom.basic",
"order.min_max",
"algebra.group.units"
] | [
"monoid",
"monotone.map_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : α) : with_top α) = 1 | rfl | lemma | with_top.coe_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_one {a : α} : (a : with_top α) = 1 ↔ a = 1 | coe_eq_coe | lemma | with_top.coe_eq_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untop_one : (1 : with_top α).untop coe_ne_top = 1 | rfl | lemma | with_top.untop_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untop_one' (d : α) : (1 : with_top α).untop' d = 1 | rfl | lemma | with_top.untop_one' | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_coe [has_le α] {a : α} : 1 ≤ (a : with_top α) ↔ 1 ≤ a | coe_le_coe | lemma | with_top.one_le_coe | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_one [has_le α] {a : α} : (a : with_top α) ≤ 1 ↔ a ≤ 1 | coe_le_coe | lemma | with_top.coe_le_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_coe [has_lt α] {a : α} : 1 < (a : with_top α) ↔ 1 < a | coe_lt_coe | lemma | with_top.one_lt_coe | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lt_one [has_lt α] {a : α} : (a : with_top α) < 1 ↔ a < 1 | coe_lt_coe | lemma | with_top.coe_lt_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one {β} (f : α → β) :
(1 : with_top α).map f = (f 1 : with_top β) | rfl | lemma | with_top.map_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"map_one",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_coe {a : α} : 1 = (a : with_top α) ↔ a = 1 | trans eq_comm coe_eq_one | theorem | with_top.one_eq_coe | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_ne_one : ⊤ ≠ (1 : with_top α) | theorem | with_top.top_ne_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | ||
one_ne_top : (1 : with_top α) ≠ ⊤ | theorem | with_top.one_ne_top | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | ||
coe_add : ((x + y : α) : with_top α) = x + y | rfl | lemma | with_top.coe_add | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bit0 : ((bit0 x : α) : with_top α) = bit0 x | rfl | lemma | with_top.coe_bit0 | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bit1 [has_one α] {a : α} : ((bit1 a : α) : with_top α) = bit1 a | rfl | lemma | with_top.coe_bit1 | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_add (a : with_top α) : ⊤ + a = ⊤ | rfl | lemma | with_top.top_add | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"top_add",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_top (a : with_top α) : a + ⊤ = ⊤ | by cases a; refl | lemma | with_top.add_top | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_top",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_top : a + b = ⊤ ↔ a = ⊤ ∨ b = ⊤ | by cases a; cases b; simp [none_eq_top, some_eq_coe, ←with_top.coe_add] | lemma | with_top.add_eq_top | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_ne_top : a + b ≠ ⊤ ↔ a ≠ ⊤ ∧ b ≠ ⊤ | add_eq_top.not.trans not_or_distrib | lemma | with_top.add_ne_top | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_top [has_lt α] {a b : with_top α} : a + b < ⊤ ↔ a < ⊤ ∧ b < ⊤ | by simp_rw [with_top.lt_top_iff_ne_top, add_ne_top] | lemma | with_top.add_lt_top | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top",
"with_top.lt_top_iff_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_coe : ∀ {a b : with_top α} {c : α},
a + b = c ↔ ∃ (a' b' : α), ↑a' = a ∧ ↑b' = b ∧ a' + b' = c | | none b c := by simp [none_eq_top]
| (some a) none c := by simp [none_eq_top]
| (some a) (some b) c :=
by simp only [some_eq_coe, ← coe_add, coe_eq_coe, exists_and_distrib_left, exists_eq_left] | lemma | with_top.add_eq_coe | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"exists_and_distrib_left",
"exists_eq_left",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_coe_eq_top_iff {x : with_top α} {y : α} : x + y = ⊤ ↔ x = ⊤ | by { induction x using with_top.rec_top_coe; simp [← coe_add] } | lemma | with_top.add_coe_eq_top_iff | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top",
"with_top.rec_top_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_eq_top_iff {y : with_top α} : ↑x + y = ⊤ ↔ y = ⊤ | by { induction y using with_top.rec_top_coe; simp [← coe_add] } | lemma | with_top.coe_add_eq_top_iff | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top",
"with_top.rec_top_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_add_le [has_le α] [covariant_class α α (+) (≤)] :
covariant_class (with_top α) (with_top α) (+) (≤) | ⟨λ a b c h, begin
cases a; cases c; try { exact le_top },
rcases le_coe_iff.1 h with ⟨b, rfl, h'⟩,
exact coe_le_coe.2 (add_le_add_left (coe_le_coe.1 h) _)
end⟩ | instance | with_top.covariant_class_add_le | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"le_top",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_swap_add_le [has_le α] [covariant_class α α (swap (+)) (≤)] :
covariant_class (with_top α) (with_top α) (swap (+)) (≤) | ⟨λ a b c h, begin
cases a; cases c; try { exact le_top },
rcases le_coe_iff.1 h with ⟨b, rfl, h'⟩,
exact coe_le_coe.2 (add_le_add_right (coe_le_coe.1 h) _)
end⟩ | instance | with_top.covariant_class_swap_add_le | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"le_top",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_add_lt [has_lt α] [contravariant_class α α (+) (<)] :
contravariant_class (with_top α) (with_top α) (+) (<) | ⟨λ a b c h, begin
induction a using with_top.rec_top_coe, { exact (not_none_lt _ h).elim },
induction b using with_top.rec_top_coe, { exact (not_none_lt _ h).elim },
induction c using with_top.rec_top_coe,
{ exact coe_lt_top _ },
{ exact coe_lt_coe.2 (lt_of_add_lt_add_left $ coe_lt_coe.1 h) }
end⟩ | instance | with_top.contravariant_class_add_lt | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"with_top",
"with_top.rec_top_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_swap_add_lt [has_lt α] [contravariant_class α α (swap (+)) (<)] :
contravariant_class (with_top α) (with_top α) (swap (+)) (<) | ⟨λ a b c h, begin
cases a; cases b; try { exact (not_none_lt _ h).elim },
cases c,
{ exact coe_lt_top _ },
{ exact coe_lt_coe.2 (lt_of_add_lt_add_right $ coe_lt_coe.1 h) }
end⟩ | instance | with_top.contravariant_class_swap_add_lt | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_add_le_add_left [has_le α] [contravariant_class α α (+) (≤)] (ha : a ≠ ⊤)
(h : a + b ≤ a + c) : b ≤ c | begin
lift a to α using ha,
induction c using with_top.rec_top_coe, { exact le_top },
induction b using with_top.rec_top_coe, { exact (not_top_le_coe _ h).elim },
simp only [← coe_add, coe_le_coe] at h ⊢,
exact le_of_add_le_add_left h
end | lemma | with_top.le_of_add_le_add_left | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"le_top",
"lift",
"with_top.rec_top_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_add_le_add_right [has_le α] [contravariant_class α α (swap (+)) (≤)]
(ha : a ≠ ⊤) (h : b + a ≤ c + a) : b ≤ c | begin
lift a to α using ha,
cases c,
{ exact le_top },
cases b,
{ exact (not_top_le_coe _ h).elim },
{ exact coe_le_coe.2 (le_of_add_le_add_right $ coe_le_coe.1 h) }
end | lemma | with_top.le_of_add_le_add_right | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"le_top",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_left [has_lt α] [covariant_class α α (+) (<)] (ha : a ≠ ⊤) (h : b < c) :
a + b < a + c | begin
lift a to α using ha,
rcases lt_iff_exists_coe.1 h with ⟨b, rfl, h'⟩,
cases c,
{ exact coe_lt_top _ },
{ exact coe_lt_coe.2 (add_lt_add_left (coe_lt_coe.1 h) _) }
end | lemma | with_top.add_lt_add_left | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_right [has_lt α] [covariant_class α α (swap (+)) (<)]
(ha : a ≠ ⊤) (h : b < c) :
b + a < c + a | begin
lift a to α using ha,
rcases lt_iff_exists_coe.1 h with ⟨b, rfl, h'⟩,
cases c,
{ exact coe_lt_top _ },
{ exact coe_lt_coe.2 (add_lt_add_right (coe_lt_coe.1 h) _) }
end | lemma | with_top.add_lt_add_right | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_add_iff_left [has_le α] [covariant_class α α (+) (≤)]
[contravariant_class α α (+) (≤)]
(ha : a ≠ ⊤) : a + b ≤ a + c ↔ b ≤ c | ⟨with_top.le_of_add_le_add_left ha, λ h, add_le_add_left h a⟩ | lemma | with_top.add_le_add_iff_left | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_add_iff_right [has_le α] [covariant_class α α (swap (+)) (≤)]
[contravariant_class α α (swap (+)) (≤)] (ha : a ≠ ⊤) : b + a ≤ c + a ↔ b ≤ c | ⟨with_top.le_of_add_le_add_right ha, λ h, add_le_add_right h a⟩ | lemma | with_top.add_le_add_iff_right | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_iff_left [has_lt α] [covariant_class α α (+) (<)]
[contravariant_class α α (+) (<)] (ha : a ≠ ⊤) : a + b < a + c ↔ b < c | ⟨lt_of_add_lt_add_left, with_top.add_lt_add_left ha⟩ | lemma | with_top.add_lt_add_iff_left | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"covariant_class",
"with_top.add_lt_add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_iff_right [has_lt α] [covariant_class α α (swap (+)) (<)]
[contravariant_class α α (swap (+)) (<)] (ha : a ≠ ⊤) : b + a < c + a ↔ b < c | ⟨lt_of_add_lt_add_right, with_top.add_lt_add_right ha⟩ | lemma | with_top.add_lt_add_iff_right | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"covariant_class",
"with_top.add_lt_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_of_le_of_lt [preorder α] [covariant_class α α (+) (<)]
[covariant_class α α (swap (+)) (≤)] (ha : a ≠ ⊤) (hab : a ≤ b) (hcd : c < d) : a + c < b + d | (with_top.add_lt_add_left ha hcd).trans_le $ add_le_add_right hab _ | lemma | with_top.add_lt_add_of_le_of_lt | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"with_top.add_lt_add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_of_lt_of_le [preorder α] [covariant_class α α (+) (≤)]
[covariant_class α α (swap (+)) (<)] (hc : c ≠ ⊤) (hab : a < b) (hcd : c ≤ d) : a + c < b + d | (with_top.add_lt_add_right hc hab).trans_le $ add_le_add_left hcd _ | lemma | with_top.add_lt_add_of_lt_of_le | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"with_top.add_lt_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add {F} [has_add β] [add_hom_class F α β] (f : F) (a b : with_top α) :
(a + b).map f = a.map f + b.map f | begin
induction a using with_top.rec_top_coe,
{ exact (top_add _).symm },
{ induction b using with_top.rec_top_coe,
{ exact (add_top _).symm },
{ rw [map_coe, map_coe, ← coe_add, ← coe_add, ← map_add],
refl } },
end | lemma | with_top.map_add | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_hom_class",
"add_top",
"top_add",
"with_top",
"with_top.rec_top_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat [add_monoid_with_one α] (n : ℕ) : ((n : α) : with_top α) = n | rfl | lemma | with_top.coe_nat | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_monoid_with_one",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_ne_top [add_monoid_with_one α] (n : ℕ) : (n : with_top α) ≠ ⊤ | coe_ne_top | lemma | with_top.nat_ne_top | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_monoid_with_one",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_ne_nat [add_monoid_with_one α] (n : ℕ) : (⊤ : with_top α) ≠ n | top_ne_coe | lemma | with_top.top_ne_nat | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_monoid_with_one",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_hom [add_monoid α] : α →+ with_top α | ⟨coe, rfl, λ _ _, rfl⟩ | def | with_top.coe_add_hom | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_monoid",
"with_top"
] | Coercion from `α` to `with_top α` as an `add_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coe_add_hom [add_monoid α] : ⇑(coe_add_hom : α →+ with_top α) = coe | rfl | lemma | with_top.coe_coe_add_hom | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_monoid",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_top [ordered_add_comm_monoid α] : (0 : with_top α) < ⊤ | coe_lt_top 0 | lemma | with_top.zero_lt_top | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"ordered_add_comm_monoid",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_coe [ordered_add_comm_monoid α] (a : α) :
(0 : with_top α) < a ↔ 0 < a | coe_lt_coe | lemma | with_top.zero_lt_coe | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"ordered_add_comm_monoid",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.one_hom.with_top_map {M N : Type*} [has_one M] [has_one N] (f : one_hom M N) :
one_hom (with_top M) (with_top N) | { to_fun := with_top.map f,
map_one' := by rw [with_top.map_one, map_one, coe_one] } | def | one_hom.with_top_map | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"map_one",
"one_hom",
"with_top",
"with_top.map",
"with_top.map_one"
] | A version of `with_top.map` for `one_hom`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.add_hom.with_top_map
{M N : Type*} [has_add M] [has_add N] (f : add_hom M N) :
add_hom (with_top M) (with_top N) | { to_fun := with_top.map f,
map_add' := with_top.map_add f } | def | add_hom.with_top_map | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_hom",
"with_top",
"with_top.map",
"with_top.map_add"
] | A version of `with_top.map` for `add_hom`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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